QUADRATIC PERTURBATIONS OF A CLASS OF QUADRATIC REVERSIBLE LOTKA–VOLTERRA SYSTEMS

2013 ◽  
Vol 23 (08) ◽  
pp. 1350137
Author(s):  
YI SHAO ◽  
A. CHUNXIANG
Keyword(s):  
Limit Cycles ◽  
Positive Answer ◽  
Abelian Integrals ◽  
Volterra System ◽  
Bifurcation Of Limit Cycles ◽  
Number Of Zeros ◽  
Period Annulus ◽  
Volterra Systems

This paper is concerned with the bifurcation of limit cycles of a class of quadratic reversible Lotka–Volterra system [Formula: see text] with b = -1/3. By using the Chebyshev criterion to study the number of zeros of Abelian integrals, we prove that this system has at most two limit cycles produced from the period annulus around the center under quadratic perturbations, which provide a positive answer for a case of the conjecture proposed by S. Gautier et al.

scholarly journals ON THE NUMBER OF LIMIT CYCLES IN PERTURBATIONS OF A QUADRATIC REVERSIBLE CENTER

2012 ◽  
Vol 92 (3) ◽  
pp. 409-423
Author(s):  
JUANJUAN WU ◽  
LINPING PENG ◽  
CUIPING LI
Keyword(s):  
Limit Cycles ◽  
Abelian Integral ◽  
Reversible System ◽  
Linear Estimate ◽  
Bifurcation Of Limit Cycles ◽  
Arbitrary Degree ◽  
Number Of Zeros ◽  
Period Annulus

AbstractThis paper is concerned with the bifurcation of limit cycles from a quadratic reversible system under polynomial perturbations. It is proved that the cyclicity of the period annulus is two, and also a linear estimate of the number of zeros of the Abelian integral for the system under polynomial perturbations of arbitrary degreenis given.

Bifurcation of Limit Cycles from the Center of a Family of Cubic Polynomial Vector Fields

2018 ◽  
Vol 28 (05) ◽  
pp. 1850063 ◽  
Author(s):  
Shiyou Sui ◽  
Liqin Zhao
Keyword(s):  
Limit Cycles ◽  
Vector Fields ◽  
Abelian Integral ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Bifurcation Of Limit Cycles ◽  
First Order ◽  
Number Of Zeros ◽  
Period Annulus ◽  
Cubic Polynomial

In this paper, we consider the number of zeros of Abelian integral for the system [Formula: see text], [Formula: see text], where [Formula: see text], [Formula: see text], and [Formula: see text] are arbitrary polynomials of degree [Formula: see text]. We obtain that [Formula: see text] if [Formula: see text] and [Formula: see text] if [Formula: see text], where [Formula: see text] is the maximum number of limit cycles bifurcating from the period annulus up to the first order in [Formula: see text]. So, the bounds for [Formula: see text] or [Formula: see text], [Formula: see text], [Formula: see text] are exact.

Number of Limit Cycles from a Class of Perturbed Piecewise Polynomial Systems

2021 ◽  
Vol 31 (09) ◽  
pp. 2150123
Author(s):  
Xiaoyan Chen ◽  
Maoan Han
Keyword(s):  
Limit Cycles ◽  
Polynomial Systems ◽  
Melnikov Function ◽  
Unperturbed System ◽  
Piecewise Polynomial ◽  
First Order ◽  
Number Of Zeros ◽  
Period Annulus

In this paper, we study Poincaré bifurcation of a class of piecewise polynomial systems, whose unperturbed system has a period annulus together with two invariant lines. The main concerns are the number of zeros of the first order Melnikov function and the estimation of the number of limit cycles which bifurcate from the period annulus under piecewise polynomial perturbations of degree [Formula: see text].

The Cyclicity of Period Annulus of Degenerate Quadratic Hamiltonian Systems with Polycycles S(2) or S(3) Under Perturbations of Piecewise Smooth Polynomials

2020 ◽  
Vol 30 (15) ◽  
pp. 2050230
Author(s):  
Jiaxin Wang ◽  
Liqin Zhao
Keyword(s):  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Upper Bounds ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
Bifurcation Of Limit Cycles ◽  
First Order ◽  
Period Annulus ◽  
Quadratic Hamiltonian

In this paper, by using Picard–Fuchs equations and Chebyshev criterion, we study the bifurcation of limit cycles for degenerate quadratic Hamilton systems with polycycles [Formula: see text] or [Formula: see text] under the perturbations of piecewise smooth polynomials with degree [Formula: see text]. Roughly speaking, for [Formula: see text], a polycycle [Formula: see text] is cyclically ordered collection of [Formula: see text] saddles together with orbits connecting them in specified order. The discontinuity is on the line [Formula: see text]. If the first order Melnikov function is not equal to zero identically, it is proved that the upper bounds of the number of limit cycles bifurcating from each of the period annuli with the boundary [Formula: see text] and [Formula: see text] are respectively [Formula: see text] and [Formula: see text] (taking into account the multiplicity).

Bifurcation of Limit Cycles in a Near-Hamiltonian System with a Cusp of Order Two and a Saddle

2016 ◽  
Vol 26 (11) ◽  
pp. 1650180 ◽  
Author(s):  
Ali Bakhshalizadeh ◽  
Hamid R. Z. Zangeneh ◽  
Rasool Kazemi
Keyword(s):  
Asymptotic Expansion ◽  
Hamiltonian System ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Upper Bound ◽  
Melnikov Function ◽  
Bifurcation Of Limit Cycles ◽  
Heteroclinic Loop ◽  
First Order ◽  
Period Annulus

In this paper, the asymptotic expansion of first-order Melnikov function of a heteroclinic loop connecting a cusp of order two and a hyperbolic saddle for a planar near-Hamiltonian system is given. Next, we consider the limit cycle bifurcations of a hyper-elliptic Liénard system with this kind of heteroclinic loop and study the least upper bound of limit cycles bifurcated from the period annulus inside the heteroclinic loop, from the heteroclinic loop itself and the center. We find that at most three limit cycles can be bifurcated from the period annulus, also we present different distributions of bifurcated limit cycles.

Inhomogeneous Fuchs equations and the limit cycles in a class of near-integrable quadratic systems

1997 ◽  
Vol 127 (6) ◽  
pp. 1207-1217 ◽  
Author(s):  
Iliya D. Iliev
Keyword(s):  
Limit Cycles ◽  
Displacement Function ◽  
Quadratic System ◽  
Degenerate Case ◽  
Second Variation ◽  
Centre Manifold ◽  
Bifurcation Of Limit Cycles ◽  
Quadratic Systems ◽  
Period Annulus ◽  
The Second Variation

We study the bifurcation of limit cycles in general quadratic perturbations of the particular quadratic system which represents one of two codimension-five components in the intersection of two strata in the centre manifold, and Q4. The study of limit cycles for this degenerate case requires us to investigate not the first but the second variation M2 of the displacement function. We prove that up to three limit cycles can emerge from the period annulus surrounding the centre. This implies that the cyclicity of period annuli of nearby systems in and Q4 is at most three as well. Our approach relies upon the possibility of deriving appropriate Picard–Fuchs equations satisfied by the four independent integrals included in M2.

ON THE NUMBER OF ZEROS OF ABELIAN INTEGRAL FOR A CUBIC ISOCHRONOUS CENTER

2012 ◽  
Vol 22 (01) ◽  
pp. 1250016 ◽  
Author(s):  
KUILIN WU ◽  
YUNLIN ZHAO
Keyword(s):  
Periodic Orbits ◽  
Limit Cycles ◽  
Upper Bound ◽  
Abelian Integral ◽  
Isochronous Center ◽  
Number Of Zeros ◽  
Period Annulus

In this paper, we study the number of limit cycles that bifurcate from the periodic orbits of a cubic reversible isochronous center under cubic perturbations. It is proved that in this situation the least upper bound for the number of zeros (taking into account the multiplicity) of the Abelian integral associated with the system is equal to four. Moreover, for each k = 0, 1, …, 4, there are perturbations that give rise to exactly k limit cycles bifurcating from the period annulus.

Bifurcation of Limit Cycles from the Center of a Quintic System via the Averaging Method

2017 ◽  
Vol 27 (05) ◽  
pp. 1750072 ◽  
Author(s):  
Bo Huang
Keyword(s):  
Limit Cycles ◽  
Upper Bound ◽  
Averaging Method ◽  
Unperturbed System ◽  
Homogeneous Polynomials ◽  
Bifurcation Of Limit Cycles ◽  
First Order ◽  
Period Annulus

This paper deals with the bifurcation of limit cycles for a quintic system with one center. Using the averaging method, we explain how limit cycles can bifurcate from the periodic annulus around the center of the considered system by adding perturbed terms which are the sum of homogeneous polynomials of degree [Formula: see text] for [Formula: see text]. We show that up to first-order averaging, at most five limit cycles can bifurcate from the period annulus of the unperturbed system for [Formula: see text], at most [Formula: see text] limit cycles can bifurcate from the periodic annulus of the unperturbed system for any [Formula: see text], and the upper bound is sharp for [Formula: see text] and for [Formula: see text].

Bifurcation of Limit Cycles from a Polynomial Degenerate Center

2010 ◽  
Vol 10 (3) ◽  
Author(s):  
Adriana Buică ◽  
Jaume Giné ◽  
Jaume Llibre
Keyword(s):  
Limit Cycles ◽  
Upper Bounds ◽  
Bifurcation Of Limit Cycles ◽  
Period Annulus ◽  
Melnikov Functions

AbstractUsing Melnikov functions at any order, we provide upper bounds for the maximum number of limit cycles bifurcating from the period annulus of the degenerate center ẋ = −y((x

CYCLICITY OF PERIOD ANNULUS OF A QUADRATIC REVERSIBLE SYSTEM WITH DEGENERATE CRITICAL POINT

2013 ◽  
Vol 23 (06) ◽  
pp. 1350106 ◽  
Author(s):  
HAIHUA LIANG ◽  
JIANFENG HUANG
Keyword(s):  
Critical Point ◽  
Limit Cycles ◽  
Upper Bound ◽  
Outer Boundary ◽  
Reversible System ◽  
Bifurcation Of Limit Cycles ◽  
Period Annulus ◽  
Degenerate Critical Point

This paper is concerned with the bifurcation of limit cycles from the period annulus of a quadratic reversible system. The outer boundary of the period annulus contains a degenerate critical point. The exact upper bound of the number of limit cycles is given. Our result shows that the conjecture on the cyclicity of (r4) system is correct.

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